1. ## Intergration by substitution

Do do you this one please

2. $\displaystyle \displaystyle \int\frac{1+x}{1+x^2}dx=\int\frac{1}{1+x^2}dx+\int \frac{x}{1+x^2}dx=$
$\displaystyle \displaystyle =\arctan x+\frac{1}{2}\int\frac{2x}{1+x^2}dx=\arctan x+\frac{1}{2}\ln (1+x^2)+C$

3. Originally Posted by camherokid
Do do you this one please
$\displaystyle \int \frac {1 + x}{1 + x^2}~dx = \int \frac {1}{1 + x^2}~dx + \int \frac {x}{1 + x^2}~dx$

$\displaystyle = \arctan x + \int \frac {x}{1 + x^2}~dx$

We proceed by substitution for the second integral:

Let $\displaystyle u = 1 + x^2$

$\displaystyle \Rightarrow du = 2x ~dx$

$\displaystyle \Rightarrow \frac {1}{2} du = x ~dx$

$\displaystyle \Rightarrow \int \frac {1 + x}{1 + x^2}~dx = \arctan x + \frac {1}{2} \int \frac {1}{u} ~du$

$\displaystyle = \arctan x + \frac {1}{2} \ln u + C$

$\displaystyle = \boxed { \arctan x + \frac {1}{2} \ln \left( 1 + x^2 \right) + C }$

EDIT: You're too fast for me red_dog!

4. $\displaystyle f(x)=\tan ^3x$ is an odd function and $\displaystyle \displaystyle \left[-\frac{\pi}{6},\frac{\pi}{6}\right]$ is a symetric interval, so the integral is $\displaystyle 0$.

5. I will do the indefinite integral for the second, i leave plugging in the limits to you

$\displaystyle \int \tan^3 x ~dx = \int \tan^2 x \tan x ~dx$

$\displaystyle = \int \left( \sec^2 x - 1 \right) \tan x ~dx$

$\displaystyle = \int \left( \sec^2 x \tan x - \tan x \right)~dx$

$\displaystyle = \int \sec^2 x \tan x ~dx - \int \tan x ~dx$

$\displaystyle = \int \sec^2 x \tan x ~dx - \ln | \sec x | + C$

We proceed by substitution for the first integral:

Let $\displaystyle u = \tan x$

$\displaystyle \Rightarrow du = \sec^2 x ~dx$

$\displaystyle \Rightarrow \int \tan^3 x ~dx = \int u ~du - \ln | \sec x | + C$

$\displaystyle = \frac {1}{2} u^2 - \ln | \sec x | + C$

$\displaystyle = \boxed { \frac {1}{2} \tan^2 x - \ln | \sec x | + C }$

6. Originally Posted by red_dog
$\displaystyle f(x)=\tan ^3x$ is an odd function and $\displaystyle \displaystyle \left[-\frac{\pi}{6},\frac{\pi}{6}\right]$ is a symetric interval, so the integral is $\displaystyle 0$.
Ah, i never noticed that (never paid attention to the limits, i planned on finding the indefinate integral from the start). That's true, the integral is 0. However, if what you actually wanted was the area, you would find $\displaystyle 2 \int_{0}^{ \frac { \pi}{6}} \tan^3 x~dx$

7. Another possible solution, similar to Jhevon's is:

$\displaystyle \int{tan^{3}(x)}dx=\int{(sec^{2}(x)-1)tan(x)}dx=sec^{2}(x)tan(x)-tan(x)$

$\displaystyle \int{sec^{2}(x)tan(x)}dx$

Now, let $\displaystyle u=sec(x), \;\ du=sec(x)tan(x)dx$

$\displaystyle \int{u}du=\frac{1}{2}u^{2}=\frac{1}{2}sec^{2}(x)$

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

$\displaystyle \int{tan(x)}dx$

$\displaystyle \int\frac{sin(x)}{cos(x)}dx$

Let $\displaystyle u=cos(x), \;\ -du=sin(x)dx$

$\displaystyle -\int\frac{1}{u}du=ln(u)=-ln(cos(x))$

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

So, we have $\displaystyle \boxed{\frac{1}{2}sec^{2}(x)+ln(cos(x))+C}$

$\displaystyle \frac{1}{2}tan^{2}(x)-ln(sec(x))+\frac{1}{2}=\frac{1}{2}sec^{2}(x)+ln(co s(x))$

8. Originally Posted by red_dog
$\displaystyle f(x)=\tan ^3x$ is an odd function and $\displaystyle \displaystyle \left[-\frac{\pi}{6},\frac{\pi}{6}\right]$ is a symetric interval, so the integral is $\displaystyle 0$.
This is the simplest method of all.
The integral of an odd function on a symmetric interval is always zero